Section23.4Generalizing Moebius

There is a more serious side as well, though, and this is our key to arithmetic functions. We will now turn to algebra again, with a goal of generalizing the Moebius result.

Subsection23.4.1The monoid of arithmetic functions

Definition23.4.1
A commutative monoid is a set with multiplication (an operation) that has an identity, is associative and commutative.

You can think of a commutative monoid as an Abelian group without inverses. (That means it's not a group.)

Can you think of other commutative monoids? What sets have an operation and an identity, but no inverse?

Subsection23.4.2Bringing in group structure

Let's get deeper in the algebraic structure behind the \(\star\) operation. Remember, \(f\star g\) is defined by \[(f\star g)(n)=\sum_{de=n}f(d)g(e)\; .\]

This structure is so neat is because it actually allows us to generalize the idea behind the Moebius function!

Notice that means exactly that the Moebius function \(\mu\) is \(\mu=u^{-1}\). In general, we can also say that \[f\star f^{-1}=I=f^{-1}\star f\] There is another, more theoretical, implication.

Try to figure out what the inverse of \(N\) or \(\phi\) is with paper and pencil.

Subsection23.4.3More dividends from structure

This new way of looking at things yields a slew of information about arithmetic functions which will yield dividends about analysis/calculus (no, we haven't forgotten that!) immediately.

We promised the following corollary at the beginning of this chapter.